Bayesian neural networks for predicting uncertainty in full-field material response

TL;DR

The work addresses the challenge of predicting high-dimensional full-field stress distributions while quantifying predictive uncertainty in solid mechanics. It introduces a CNN-based Bayesian surrogate (a modified U-net) and compares three inference schemes—Hamiltonian Monte Carlo (HMC), Monte Carlo Dropout (MCD), and Bayes by Backprop (BBB)—across fiber-reinforced and polycrystalline microstructures. The results show that HMC achieves the strongest accuracy and consistent epistemic uncertainty estimates, BBB provides a robust, computationally efficient alternative with reliable UQ, while MCD is highly configuration-dependent and less reliable for spatial uncertainty. The approach demonstrates that Bayesian neural networks can deliver accurate mean predictions and spatially resolved uncertainty, enabling more trustworthy surrogates for FE-based material simulations with potential broad applicability beyond stress prediction. Practically, this framework supports efficient uncertainty-aware screening of microstructures and can be extended to handle more complex noise, higher dimensions, and parallelized inference strategies.

Abstract

Stress and material deformation field predictions are among the most important tasks in computational mechanics. These predictions are typically made by solving the governing equations of continuum mechanics using finite element analysis, which can become computationally prohibitive considering complex microstructures and material behaviors. Machine learning (ML) methods offer potentially cost effective surrogates for these applications. However, existing ML surrogates are either limited to low-dimensional problems and/or do not provide uncertainty estimates in the predictions. This work proposes an ML surrogate framework for stress field prediction and uncertainty quantification for diverse materials microstructures. A modified Bayesian U-net architecture is employed to provide a data-driven image-to-image mapping from initial microstructure to stress field with prediction (epistemic) uncertainty estimates. The Bayesian posterior distributions for the U-net parameters are estimated using three state-of-the-art inference algorithms: the posterior sampling-based Hamiltonian Monte Carlo method and two variational approaches, the Monte-Carlo Dropout method and the Bayes by Backprop algorithm. A systematic comparison of the predictive accuracy and uncertainty estimates for these methods is performed for a fiber reinforced composite material and polycrystalline microstructure application. It is shown that the proposed methods yield predictions of high accuracy compared to the FEA solution, while uncertainty estimates depend on the inference approach. Generally, the Hamiltonian Monte Carlo and Bayes by Backprop methods provide consistent uncertainty estimates. Uncertainty estimates from Monte Carlo Dropout, on the other hand, are more difficult to interpret and depend strongly on the method's design.

Figures (9)

• Figure 1: Schematic representation of the surrogate modeling and uncertainty quantification scheme. The U-net architecture incorporates dropout layers for the MCD technique or probabilistic filter parameters for the BBB and HMC methods. For predictions, the dropout layers or the Bayesian parameters are sampled to generate multiple stress predictions. The mean prediction is obtained averaging the resulting fields and model uncertainty is estimated by computing the standard deviation of the ensemble predictions.
• Figure 2: Stress field prediction using a standard deterministic U-net for one representative microstructure in the fiber-reinforced composite test dataset. Absolute error refers to the absolute value of the difference between the stresses predicted by FE and U-net.
• Figure 3: Trace plots for two representative NN parameters for a total of $N_s=1000$ HMC samples.
• Figure 4: Mean stress field predictions, associated uncertainties, and absolute errors for the mean prediction from a Bayesian U-net for one representative microstructure in the fiber-reinforced composite test dataset: (a) HMC implementation, (b) BBB implementation, (c) MCD implementation.
• Figure 5: Estimated epistemic uncertainty fields using the MCD algorithm with different combinations of dropout layer locations and dropout rates for a representative microstructure in the fiber-reinforced composite test dataset.